Implications

We have the following indirect implication of form equivalence classes:

7 \(\Rightarrow\) 376
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
7 \(\Rightarrow\) 9 On the existence of large sets of Dedekind cardinals, Tarski, A. 1965, Notices Amer. Math. Soc.
The Axiom of Choice, Jech, 1973b, page 161 problem 11.6
9 \(\Rightarrow\) 376 clear

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number Statement
7:

There is no infinite decreasing sequence of cardinals.

9:

Finite \(\Leftrightarrow\) Dedekind finite: \(W_{\aleph_{0}}\) Jech [1973b]: \(E(I,IV)\) Howard/Yorke [1989]): Every Dedekind finite set is finite.

376:

Restricted Kinna Wagner Principle:  For every infinite set \(X\) there is an infinite subset \(Y\) of \(X\) and a function \(f\) such that for every \(z\subseteq Y\), if \(|z| \ge 2\) then \(f(z)\) is a non-empty proper subset of \(z\).

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